The scalar of curvature power spectrum in
WI [1]}, [2]}, [3]}, [4]}
has strong dependence on the dissipation ratio \(Q\)  and on the
temperature over Hubble parameter ratioStrictly speaking
\(T/H\)  can also be considered as a function of \(Q\) , once we use the
CMB amplitude value to constrain the scalar power spectrum. In this
case, \(Q\)  can be considered the relevant quantity parameterizing the
WI dynamics., \(T/H\) . It is then useful to look at the dynamical
behavior of these quantities during inflation. In particular, the
analysis of their behavior is important to determine regimes where the
spectral tilt of the scalar of curvature power spectrum increases
(leading potentially to a blue-tilted spectrum) or decreases (and that
can lead to a red-tilted spectrum). Such an analysis is particularly
relevant in modeling WI models and constraining them with the
observations [5]}. Furthermore, since \(Q\)  appears
explicitly in the slow-roll parameters of WI, the behavior of \(Q\) 
during WI is important for determining regimes where the dynamics is
consistent and can have a graceful exit [6]}. In the
case of DBIWI, since the slow-roll coefficients also depend on the
sound speed \(c_s\) , hence, for completeness, we also study its behavior
here. During slow-roll we find that the dissipation ratio \(Q\) , the
temperature over Hubble parameter \(T/H\)  and the sound speed \(c_s\)  in
DBIWI evolve with the number of efolds \(N\) , respectively, as
\(\frac{d \ln Q}{dN} &=& \frac{c_s \left\lbrace \left[4 + c(1+c_s^2)\right]\epsilon _V - c (1 + c_s^2) \eta _V - (2 c (1 - c_s^2)+ 4 p) \kappa _V\right\rbrace }{4 - c + 4 Q + c c_s^2 Q},\\ \frac{d \ln (T/H)}{dN} &=& \frac{c_s \left\lbrace \left(6 + c_s^2 + 5 Q +c (-1 + c_s^2 Q)\right) \epsilon _V - (1 + c_s^2) (1 + Q) \eta _V +\left[p(c_{s}^{2}Q-1) - 2(1+Q) (1-c_{s}^{2}) \right] \kappa _V\right\rbrace }{(1 + Q) (4 - c + 4 Q + c c_s^2 Q)}, \nonumber \\\\ \frac{d \ln c_s}{dN} &=& \frac{ c_s (1 - c_s^2) \left\lbrace (-4 + c + cQ) \epsilon _V - (-4 + c) (1 + Q) \eta _V - 2 \left[4 - c + (4 + c +2 p) Q\right] \kappa _V\right\rbrace  }{(1 + Q) (4 - c + 4 Q + c c_s^2Q)},\) 
